In this talk we analyze the structure of shift-invariant spaces. We consider the case of a single compactly supported refinable generator. The spectral properties of a finite matrix provide useful information about the structure of the transition operator, and the assumption of global linear independence of the integer translates of the generator allows us to get results on the structure of the shift-invariant space itself. This information could be potentially useful in the construction of wavelets, sampling theory in shift invariant spaces, approximation theory, and many other topics. The extension of these results to finitely many generators and higher dimensions is in progress.